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An oracle property of the Nadaraya–Watson kernel estimator for high‐dimensional nonparametric regression
Author(s) -
Conn Daniel,
Li Gang
Publication year - 2019
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12370
Subject(s) - mathematics , estimator , covariate , nonparametric regression , curse of dimensionality , kernel regression , statistics , nonparametric statistics , regression , rate of convergence , kernel (algebra) , econometrics , combinatorics , computer science , computer network , channel (broadcasting)
The Nadaraya–Watson estimator is among the most studied nonparametric regression methods. A classical result is that its convergence rate depends on the number of covariates and deteriorates quickly as the dimension grows. This underscores the “curse of dimensionality” and has limited its use in high‐dimensional settings. In this paper, however, we show that the Nadaraya–Watson estimator has an oracle property such that when the true regression function is single‐ or multi‐index, it discovers the low‐rank dependence structure between the response and the covariates, mitigating the curse of dimensionality. Specifically, we prove that, using K ‐fold cross‐validation and a positive‐semidefinite bandwidth matrix, the Nadaraya–Watson estimator has a convergence rate that depends on the number of indices rather than on the number of covariates. This result follows by allowing the bandwidths to diverge to infinity rather than restricting them all to converge to zero at certain rates, as in previous theoretical studies.